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This new and exciting historical book tells how Euler introduced the idea of orthogonal polynomials and how he combined them with continued fractions, as well as how Brouncker's formula of 1655 can be derived from Euler's efforts in Special Functions and Orthogonal Polynomials. The most interesting applications of this work are discussed, including the great Markoff's Theorem on the Lagrange spectrum, Abel's Theorem on integration in finite terms, Chebyshev's Theory of Orthogonal Polynomials, and very recent advances in Orthogonal Polynomials on the unit circle. As continued fractions become more important again, in part due to their use in finding algorithms in approximation theory, this timely book revives the approach of Wallis, Brouncker and Euler and illustrates the continuing significance of their influence. A translation of Euler’s famous paper ‘Continued Fractions, Observation’ is included as an Addendum.Read more
- Considers the modern state of continued fractions and orthogonal polynomials from Euler's point of view, giving a full account of his work on the subject
- Outlines Brouncker's formula; Euler's discoveries of the Gamma and Beta functions; Markoff's Theorem on the Lagrange spectrum and its relation with Jean Bernoulli sequences; Brouncker's method as a solution to Fermat's question on Pell's equation
- Contains the first English translation of Euler's 'Continued Fractions, Observation', 1739, with comments relating it to Brouncker's proof
Reviews & endorsements
"A unique, fascinating book."
D.V. Feldman, University of New Hampshire for Choice MagazineSee more reviews
"The resulting book is a pleasure to read for people interested in either the topic of orthogonal polynomials and continued fractions or for historians of mathematics, and I imagine that any reader will walk away with a deeper appreciation of both."
Darren Glass, MAA Reviews
"The mathematics contained in this book is both beautiful and difficult. The author has done an admirable job of putting together historical anecdotes and excerpts from original sources with some deep and modern mathematics. The book is a pleasure to read for people interested in either orthogonal polynomials and continued fractions or the history of mathematics, and I imagine that any reader will walk away with a deeper appreciation of both."
Leonid B. Golinskii, Mathematical Reviews
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- Date Published: September 2008
- format: Hardback
- isbn: 9780521854191
- length: 496 pages
- dimensions: 241 x 163 x 31 mm
- weight: 0.86kg
- contains: 12 b/w illus. 180 exercises
- availability: Available
Table of Contents
1. Continued fractions: real numbers
2. Continued fractions: Algebra
3. Continued fractions: Analysis
4. Continued fractions: Euler
5. Continued fractions: Euler's Influence
7. Orthogonal polynomials
8. Orthogonal polynomials on the unite circle
A1. Continued fractions, Observations
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